Sailing to criterion sets for quadratic forms
Public support
Provider
Czech Science Foundation
Programme
Standard projects
Call for proposals
SGA0202600001
Main participants
Univerzita Karlova / Matematicko-fyzikální fakulta
Contest type
VS - Public tender
Contract ID
26-20514S
Alternative language
Project name in Czech
Sailing to criterion sets for quadratic forms
Annotation in Czech
Number fields, quadratic forms, and continued fractions are among the most classical and prominent objects of study in number theory. In this project, we will discover and establish novel connections between the arithmetic of number fields and the geometry of generalized continued fractions. Specifically, this will involve relating universal quadratic forms and their criterion sets, additively indecomposable integers in totally real number fields, and sails of lattices in R^n. Championed by our team, the last decade saw remarkable progress on these topics over real quadratic fields, i.e., the case n=2, and some promising, but sporadic results in higher degrees. Our principal goal is to uncover a deeper theory connecting sails, indecomposables, and criterion sets. As applications of our new theory, we aim at significant breakthroughs on Arnold's conjectures related to realizability for sails, Kitaoka’s conjecture on ranks of universal quadratic forms, and the broad hypothesis that criterion sets are not as big and incomprehensible as expected so far.
Scientific branches
R&D category
ZV - Basic research
OECD FORD - main branch
10101 - Pure mathematics
OECD FORD - secondary branch
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OECD FORD - another secondary branch
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CEP - equivalent branches <br>(according to the <a href="http://www.vyzkum.cz/storage/att/E6EF7938F0E854BAE520AC119FB22E8D/Prevodnik_oboru_Frascati.pdf">converter</a>)
BA - General mathematics
Solution timeline
Realization period - beginning
Jan 1, 2026
Realization period - end
Dec 31, 2028
Project status
Z - Beginning multi-year project
Latest support payment
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Data delivery to CEP
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data delivery code
CEP26-GA0-GA-R
Data delivery date
Apr 27, 2026
Finance
Total approved costs
11,129 thou. CZK
Public financial support
10,718 thou. CZK
Other public sources
411 thou. CZK
Non public and foreign sources
0 thou. CZK